U.S. patent number 6,391,645 [Application Number 08/854,440] was granted by the patent office on 2002-05-21 for method and apparatus for correcting ambient temperature effect in biosensors.
This patent grant is currently assigned to Bayer Corporation. Invention is credited to Dijia Huang, Brenda L. Tudor, Kin-Fai Yip.
United States Patent |
6,391,645 |
Huang , et al. |
May 21, 2002 |
Method and apparatus for correcting ambient temperature effect in
biosensors
Abstract
A method and apparatus are provided for correcting ambient
temperature effect in biosensors. An ambient temperature value is
measured. A sample is applied to the biosensors, then a current
generated in the test sample is measured. An observed analyte
concentration value is calculated from the current through a
standard response curve. The observed analyte concentration is then
modified utilizing the measured ambient temperature value to
thereby increase the accuracy of the analyte determination. The
analyte concentration value can be calculated by solving the
following equation: ##EQU1## where G.sub.1 is said observed analyte
concentration value, T.sub.2 is said measured ambient temperature
value and I1, I2, S1, and S2 are predetermined parameters.
Inventors: |
Huang; Dijia (Granger, IN),
Tudor; Brenda L. (Elkhart, IN), Yip; Kin-Fai (Elkhart,
IN) |
Assignee: |
Bayer Corporation (Elkhart,
IN)
|
Family
ID: |
25318702 |
Appl.
No.: |
08/854,440 |
Filed: |
May 12, 1997 |
Current U.S.
Class: |
436/95;
422/82.02; 436/151 |
Current CPC
Class: |
C12Q
1/006 (20130101); G01N 27/3274 (20130101); Y10T
436/144444 (20150115) |
Current International
Class: |
C12Q
1/00 (20060101); G01N 33/487 (20060101); G01N
027/12 () |
Field of
Search: |
;422/82.02
;436/95,151,152 ;435/14,25,287.1,817 ;205/777.5 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Snay; Jeffrey
Attorney, Agent or Firm: Jeffers; Jerome L.
Claims
What is claimed is:
1. A method for correcting ambient temperature effect in biosensors
comprising the steps of:
measuring an ambient temperature value;
applying a sample to the biosensors and measuring a current
generated in the test sample;
calculating an analyte concentration value utilizing said measured
ambient temperature value to thereby increase the accuracy of the
analyte determination; and
said step of calculating said analyte concentration value includes
the step of converting said measured current to an observed analyte
concentration value and calculating a corrected analyte
concentration value utilizing the equation: ##EQU6##
where G.sub.1 is said observed analyte concentration value, T.sub.2
is said measured ambient temperature value and I1, I2, S1, and S2
are set values and are experimentally determined coefficients.
2. A method for correcting ambient temperature effect in biosensors
as recited in claim 1 wherein I1, I2, S1, and S2 are experimentally
determined coefficients by sequentially providing a sample to a
plurality of concentration values and measuring a resulting current
response for each of the plurality of concentration values for each
of a plurality of temperature values; generating plots of said
observed analyte concentration values and using the following
equations:
and the generated plots to determine I1, I2, S1, and S2.
3. A method for correcting ambient temperature effect in biosensors
as recited in claim 1 wherein the analyte is glucose.
4. Apparatus for correcting ambient temperature effect in
biosensors comprising:
means for measuring an ambient temperature value;
means responsive to an applied sample to the biosensors, for
measuring a current generated in the test sample;
means for calculating an analyte concentration value utilizing said
measured ambient temperature value to thereby increase the accuracy
of the analyte determination; and
wherein said means for calculating said analyte concentration value
includes means for converting said measured current to an observed
analyte concentration value and for calculating a corrected analyte
concentration value utilizing the equation: ##EQU7##
where G.sub.1 is said observed analyte concentration value, T.sub.2
is said measured ambient temperature value and I1, I2, S1, and S2
are set values and are experimentally determined coefficients.
5. Apparatus for correcting ambient temperature effect in
biosensors as recited in claim 4 includes processor means for
performing a predefined test sequence; and,wherein said means for
measuring said ambient temperature value includes a thermistor
coupled to said processor means.
6. Apparatus for correcting ambient temperature effect in
biosensors as recited in claim 4 wherein I1, I2, S1, and S2 are
experimentally determined coefficients by sequentially providing a
sample to a plurality of concentration values and measuring a
resulting current response for each of the plurality of
concentration values for each of a plurality of temperature values;
generating plots of said observed analyte concentration values and
using the following equations:
and the generated plots to determine I1, I2, S1, and S2.
7. Apparatus for correcting ambient temperature effect in
biosensors as recited in claim 4 wherein the analyte is
glucose.
8. Apparatus for correcting ambient temperature effect in
biosensors as recited in claim 4 wherein said means responsive to
said applied sample to the biosensors, for measuring said current
generated in the test sample includes processor means coupled to
the biosensors for receiving a signal representing said current
generated in the test sample.
9. Apparatus for correcting ambient temperature effect in
biosensors as recited in claim 4 wherein said means for calculating
said analyte concentration value includes processor means coupled
to said ambient temperature measuring means and said current
measuring means and including means for solving said equation
utilizing said measured values and predetermined coefficient
values.
10. A biosensor comprising:
biosensors means for receiving a user sample;
processor means responsive to said user sample receiving means, for
measuring a current generated in the test sample;
means for measuring an ambient temperature value; and
means for calculating an analyte concentration value utilizing said
measured ambient temperature value to thereby increase the accuracy
of the analyte determination; and
wherein said processor means includes means for converting said
measured current to an observed analyte concentration value and for
calculating a corrected analyte concentration value utilizing the
equation: ##EQU8##
where G.sub.1 is said observed analyte concentration value, T.sub.2
is said measured ambient temperature value and I1, I2, S1, and S2
are set values and are experimentally determined coefficients.
11. A biosensor as recited in claim 10 wherein I1, I2, S1, and S2
are experimentally determined coefficients by sequentially
providing a sample to a plurality of concentration values and
measuring a resulting current response for each of the plurality of
concentration values for each of a plurality of temperature values,
generating plots of said observed analyte concentration values and
using the following equations:
and the generated plots to determine I1, I2, S1, and S2.
12. A biosensor as recited in claim 10 wherein said means for
measuring said ambient temperature value include a thermistor
coupled to said processor means.
Description
FIELD OF THE INVENTION
The present invention relates to a biosensor, and, more
particularly, to a new and improved method and apparatus for
correcting ambient temperature effect in biosensors.
DESCRIPTION OF THE PRIOR ART
The quantitative determination of analytes in body fluids is of
great importance in the diagnoses and maintenance of certain
physiological abnormalities. For example lactate, cholesterol and
bilirubin should be monitored in certain individuals. In
particular, the determination of glucose in body fluids is of great
importance to diabetic individuals who must frequently check the
level of glucose in their body fluids as a means of regulating the
glucose intake in their diets. While the remainder of the
disclosure herein will be directed towards the determination of
glucose, it is to be understood that the procedure and apparatus of
this invention can be used for the determination of other analytes
upon selection of the appropriate enzyme. The ideal diagnostic
device for the detection of glucose in fluids must be simple, so as
not to require a high degree of technical skill on the part of the
technician administering the test. In many cases, these tests are
administered by the patient which lends further emphasis to the
need for a test which is easy to carry out. Additionally, such a
device should be based upon elements which are sufficiently stable
to meet situations of prolonged storage.
Methods for determining analyte concentration in fluids can be
based on the electrochemical reaction between an enzyme and the
analyte specific to the enzyme and a mediator which maintains the
enzyme in its initial oxidation state. Suitable redox enzymes
include oxidases, dehydrogenases, catalase and peroxidase. For
example, in the case where glucose is the analyte, the reaction
with glucose oxidase and oxygen is represented by equation (A).
##EQU2##
In a calorimetric assay, the released hydrogen peroxide, in the
presence of a peroxidase, causes a color change in a redox
indicator which color change is proportional to the level of
glucose in the test fluid. While calorimetric tests can be made
semi-quantitative by the use of color charts for comparison of the
color change of the redox indicator with the color change obtained
using test fluids of known glucose concentration, and can be
rendered more highly quantitative by reading the result with a
spectrophotometric instrument, the results are generally not as
accurate nor are they obtained as quickly as those obtained using
an electrochemical biosensor. As used herein, the term biosensor is
intended to refer to an analytical device that responds selectively
to analytes in an appropriate sample and converts their
concentration into an electrical signal via a combination of a
biological recognition signal and a physico-chemical transducer.
Aside from its greater accuracy, a biosensor is an instrument which
generates an electrical signal directly thereby facilitating a
simplified design. Furthermore, a biosensor offers the advantage of
low material cost since a thin layer of chemicals is deposited on
the electrodes and little material is wasted.
The electron flow is then converted to the electrical signal which
directly correlates to the glucose concentration.
In the initial step of the reaction represented by equation (A),
glucose present in the test sample converts the oxidized flavin
adenine dinucleotide (FAD) center of the enzyme into its reduced
form, (FADH.sub.2).
Because these redox centers are essentially electrically insulated
within the enzyme molecule, direct electron transfer to the surface
of a conventional electrode does not occur to any measurable degree
in the absence of an unacceptably high overvoltage. An improvement
to this system involves the use of a nonphysiological redox
coupling between the electrode and the enzyme to shuttle electrons
between the (FADH.sub.2) and the electrode. This is represented by
the following scheme in which the redox coupler, typically referred
to as a mediator, is represented by M:
In this scheme, GO(FAD) represents the oxidized form of glucose
oxidase and GO(FADH.sub.2) indicates its reduced form. The
mediating species M.sub.red shuttles electrons from the reduced
enzyme to the electrode thereby oxidizing the enzyme causing its
regeneration in situ which, of course, is desirable for reasons of
economy. The main purpose for using a mediator is to reduce the
working potential of the sensor. An ideal mediator would be
re-oxidized at the electrode at a low potential under which
impurity in the chemical layer and interfering substances in the
sample would not be oxidized thereby minimizing interference.
Many compounds are useful as mediators due to their ability to
accept electrons from the reduced enzyme and transfer them to the
electrode. Among the mediators known to be useful as electron
transfer agents in analytical determinations are the substituted
benzo and naphthoquinones disclosed in U.S. Pat. No. 4,746,607; the
N-oxides, nitroso compounds, hydroxylamines and oxines specifically
disclosed in EP 0 354 441; the flavins, phenazines, phenothiazines,
indophenols, substituted 1,4-benzoquinones and indamins disclosed
in EP 0 330 517 and the phenazinium/phenoxazinium salts described
in U.S. Pat. No. 3,791,988. A comprehensive review of
electrochemical mediators of biological redox systems can be found
in Analytica Clinica Acta. 140 (1982), Pp 1-18.
Among the more venerable mediators is hexacyanoferrate, also known
as ferricyanide, which is discussed by Schlapfer et al in Clinica
Chimica Acta., 57 (1974), Pp. 283-289. In U.S. Pat. No. 4,929,545
there is disclosed the use of a soluble ferricyanide compound in
combination with a soluble ferric compound in a composition for
enzymatically determining an analyte in sample. Substituting the
iron salt of ferricyanide for oxygen in equation (A) provides:
##STR1##
since the ferricyanide is reduced to ferrocyanide by its acceptance
of electrons from the glucose oxidase enzyme.
Another way of expressing this reaction is by use of the following
equation (C):
The electrons released are directly proportional to the amount of
glucose in the test fluid and can be related thereto by measurement
of the current which is produced upon the application of a
potential thereto. Oxidation of the ferrocyanide at the anode
renews the cycle.
SUMMARY OF THE INVENTION
Important objects of the present invention are to provide a new and
improved method and apparatus for correcting ambient temperature
effect in biosensors; to provide such method and apparatus that
eliminates or minimizes the ambient temperature effect in analyte
concentration value identified by a biosensor; and to provide such
method and apparatus that overcome many of the disadvantages of
prior art arrangements.
In brief, a method and apparatus are provided for correcting
ambient temperature effect in biosensors. An ambient temperature
value is measured. A sample is applied to the biosensors, then a
current generated in the test sample is measured. An observed
analyte concentration value is calculated from the current through
a standard response curve. The observed analyte concentration is
then modified utilizing the measured ambient temperature value to
thereby increase the accuracy of the analyte determination.
In accordance with a feature of the invention, the analyte
concentration value is calculated by solving the following
equation: ##EQU3##
where G.sub.1 is said observed analyte concentration value, T.sub.2
is said measured ambient temperature value and I1, I2, S1, and S2
are predetermined parameters.
BRIEF DESCRIPTION OF THE DRAWING
The present invention together with the above and other objects and
advantages may best be understood from the following detailed
description of the preferred embodiments of the invention
illustrated in the drawings, wherein:
FIG. 1 is a block diagram representation of biosensor in accordance
with the present invention;
FIG. 2 is a flow chart illustrating logical steps performed in
accordance with the present invention of the method correcting
ambient temperature effect in biosensors by the biosensor of FIG.
1.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Having reference now to the drawings, in FIG. 1 there is shown a
block diagram representation of biosensor system designated as a
whole by the reference character 100 and arranged in accordance
with principles of the present invention. Biosensor system 100
includes a microprocessor 102 together with an associated memory
104 for storing program and user data. A meter function 106 coupled
to biosensor 108 is operatively controlled by the microprocessor
102 for recording test values, such as blood glucose test values.
An ON/OFF input at a line 110 responsive to the user ON/OFF input
operation is coupled to the microprocessor 102 for performing the
blood test sequence mode of biosensor system 100. A system features
input at a line 112 responsive to a user input operation is coupled
to the microprocessor 102 for selectively performing the system
features mode of biosensor 100. A signal input indicated at a line
120 is coupled to the microprocessor 102 providing temperature
information from a thermistor 122 in accordance with the invention.
Microprocessor 102 contains suitable programming to perform the
methods of the invention as illustrated in FIG. 2.
A display 150 is coupled to the microprocessor 102 for displaying
information to the user including test results. A battery monitor
function 160 is coupled to the microprocessor 102 for detecting a
low or dead battery condition. An alarm function 162 is coupled to
the microprocessor 102 for detecting predefined system conditions
and for generating alarm indications for the user of biosensor
system 100. A data port or communications interface 164 couples
data to and from a connected computer (not shown).
In accordance with the invention, to reduce the temperature bias,
biosensor system 100 performs a temperature correction method of
the preferred embodiment.
Referring to FIG. 2, logical steps performed in accordance with the
method for correcting ambient temperature effect in biosensors 108
by the biosensor processor 102 begin at block 200. First ambient
temperature is measured as indicated at a block 202 labeled MEASURE
INSTRUMENT TEMPERATURE T2. Then sensor current is measured as
indicated at a block 204. Next the measured current value is
converted into an analyte concentration value, such as glucose
concentration value (observed concentration), as indicated at a
block 206. Then correction for temperature effect is performed in a
final glucose concentration calculation as indicated at a block
208. The temperature corrected glucose concentration is calculated
utilizing the following equation: ##EQU4##
where G.sub.1 is said observed analyte concentration value, T.sub.2
is said measured ambient temperature value and I1, I2, S1, and S2
are predetermined parameters. This completes the sequence as
indicated at a block 210.
Amperometric biosensors 108 are known to be sensitive to
temperature. This temperature effect occurs because diffusion of
the mediator to the working electrode is temperature dependent.
Diffusion typically induces a temperature effect of 1-2% bias per
degree centigrade. Therefore temperatures as low as 10.degree. C.
would produce results with a bias of about -25% and temperatures as
high as 40.degree. C. would produce results with a bias about +25%.
The system 100 instrument provides results between 0 to 50.degree.
C. The only available temperature measurement comes from a
thermistor inside the instrument. In order to reduce the
temperature bias it was necessary to develop a temperature
correction algorithm.
The temperature effect was determined experimentally by biosensor
system 100 whole blood glucose assay over the entire glucose (50 to
600 mg/dL) and temperature range (10 to 40.degree. C.) expected to
be A encountered. Actual blood glucose readings and sample
temperatures were measured. This was done for six different sensor
108 lots. When the "compound interest" temperature correction
method was used, several lots had percent biases of -10% to -13% at
the extreme temperatures. The formula for the "compound interest"
correction method is:
where G.sub.1 is the observed glucose concentration, tc is the
temperature coefficient determined experimentally and T is the
sample temperature.
The "compound interest" algorithm did not work well because the
temperature coefficient, tc, changed with glucose concentration. A
"polynomial" correction algorithm was invented to handle the
varying temperature coefficient problem. By using a polynomial
correction algorithm, the percent bias was limited to within
+/-10%. The equation for the polynomial correction method is
described in Equation #2. The grand sum of the absolute bias for
both methods indicated that the polynomial correction method had
less overall bias. Also, at the very extreme temperatures of 2 and
49.degree. C., the polynomial correction method had lower bias
(below 13.5%) where as the compound interest method was as high as
-25%.
Therefore, the polynomial correction method provided an improvement
over the "compound interest" correction method.
After running the glucose assay at different temperatures the
current response at each temperature was calculated through the
24.degree. C. (sample temperature) standard response curve to
obtain the observed glucose concentration.
The observed glucose concentration and the sample temperature were
then used to calculate the corrected glucose concentration using
the following equation: ##EQU5##
where G.sub.1 is the observed glucose concentration, T.sub.2 is the
sample temperature and I1, I2, S1, and S2 are the predetermined
coefficients. These coefficients were determined experimentally.
See the following exemplary procedure for details.
Table 1 shows an example of the temperature correction results.
T.sub.2 is the sample temperature. G.sub.R is the reference glucose
valve. I is the measured current. G.sub.1 is the observed glucose
concentration (without temperature correction). % B is the percent
bias without temperature correction. G.sub.2 is the temperature
corrected glucose concentration. % B.sub.C is the percent bias
after temperature correction.
The data shows the percent bias before and after the correction
algorithm was applied. The algorithm and coefficients were able to
reduce the percent bias at the extreme temperatures of 10 to
40.degree. C. to within +/-7%.
EXAMPLE
TABLE 1 Temperature Correction for Lot C I1 0.17706 I2 -0.0086 S1
0.01529 S2 0.00004 Lot C T.sub.2 G.sub.R I G.sub.1 % B G.sub.2 %
B.sub.C 8.7 50 1024 38.3 -23.4% 49.1 -1.8% 8.7 100 1484 78.6 -21.4%
102.9 2.9% 8.7 200 2404 159.1 -20.5% 210.6 5.3% 8.7 400 4243 320.1
-20.0% 426.0 6.5% 8.7 600 6082 481.2 -19.8% 641.4 6.9% 16.7 50 1109
45.7 -8.6% 50.6 1.3% 16.7 100 1608 89.4 -10.6% 100.4 0.4% 16.7 200
2606 176.8 -11.6% 199.9 0.0% 16.7 400 4602 351.6 -12.1% 398.3 -0.3%
16.7 600 6598 526.4 -12.3% 597.9 -0.3% 23.9 50 1158 50.0 0.0% 50.0
0.0% 23.9 100 1729 100.0 0.0% 100.0 0.0% 23.9 200 2871 200.0 0.0%
200.0 0.0% 23.9 400 5155 400.0 0.0% 400.0 0.0% 23.9 600 7439 600.0
0.0% 600.0 0.0% 30.6 50 1212 54.7 9.5% 50.8 1.5% 30.6 100 1851
110.6 10.6% 100.8 0.8% 30.6 200 3128 222.5 11.2% 200.9 0.5% 30.6
400 5682 446.1 11.5% 401.1 0.3% 30.6 600 8236 669.8 11.6% 601.3
0.2% 38.2 50 1251 58.1 16.2% 50.4 0.8% 38.2 100 2008 124.4 24.4%
103.3 3.3% 38.2 200 3522 257.0 28.5% 209.0 4.5% 38.2 400 6550 522.1
30.5% 420.4 5.1% 38.2 600 9578 787.3 31.2% 631.8 5.3%
The following describes an exemplary procedure used for determining
the temperature correction coefficients (I.sub.1, I.sub.2, S.sub.1,
S.sub.2 in Equation 2). First venous heparinized whole blood
(.sup..about. 45% hematocrit) from a single donor was spiked close
to different glucose concentrations (values determined by the
Yellow Springs Instrument, YSI, reference method and corrected for
any known sample interferences) and tested in system 100
instruments at different environmental chamber temperatures (Table
1, e.g. samples of 50 and 400 mg/dL glucose at 8.7, 16.7, 23.9,
30.6 and 38.2.degree. C.sub.x.) The Yellow Springs Instrument and
method are described by Conrad et al., in the February 1989
"Journal of Pediatrics" Pages 281-287 and by Burmeister et al., in
"Analytical Letters", 28(4), 581-592 (1995). High relative humidity
(65 to 85%) was maintained in the chamber in order to prevent
evaporative cooling, and the sample was equilibrated to the chamber
temperature; this way the temperature effect would result only from
the chemistry. The actual sample temperature was measured for each
glucose spike. To determine the sample temperature, a 0.0005"
thermocouple was inserted into a sensor without chemistry, and
temperature data was collected every second after the blood was
added to the sensor.
TABLE 2 Lot C Actual YSI Glucose and Current Response Sample Temp.
YSI Current Slope Intercept 8.7.degree. C. 54.2 1063 8.7.degree. C.
412.5 4358 9.20 564.6 16.7.degree. C. 54.9 1148 16.7.degree. C.
414.9 4750 9.98 610.2 23.9.degree. C. 55.7 1223 23.9.degree. C. 418
5359 11.42 587.1 30.6.degree. C. 49.3 1203 30.6.degree. C. 408.4
5787 12.77 573.7 38.2.degree. C. 51.6 1275 38.2.degree. C. 418.7
6833 15.14 493.8
Next, the current response at exactly 50, 100, 200, 400, and 600
mg/dL glucose for each temperature was determined through the
curves using the slope and intercepts determined in Table 2. Using
these calculated current values the observed glucose concentration
was determined through the 24.degree. C. curve as provided in Table
3.
TABLE 3 Lot C - Current Through the YSI 50 and 400 mg/dL Curves and
the Observed Glucose mg/dL Through the 24.degree. C. Curve YSI
23.9.degree. C. Curve Sample Reference Observed Temperature
.degree. C. Glucose mg/dL Current Glucose mg/dL 8.7 50 1024 38.3
8.7 100 1484 78.6 8.7 200 2404 159.1 8.7 400 4243 320.1 8.7 600
6082 481.2 16.7 50 1109 45.7 16.7 100 1608 89.4 16.7 200 2606 176.8
16.7 400 4602 351.6 16.7 600 6598 526.4 23.9 50 1158 50.0 23.9 100
1729 100.0 23.9 200 2871 200.0 23.9 400 5155 400.0 23.9 600 7439
600.0 30.6 50 1212 57.7 30.6 100 1851 110.6 30.6 200 3128 222.5
30.6 400 5682 446.1 30.6 600 8236 669.8 38.2 50 1251 58.1 38.2 100
2008 124.4 38.2 200 3522 257.0 38.2 400 6550 522.1 38.2 600 9578
787.3
Next for each spike of blood, the observed glucose concentration
(G.sub.1) was plotted against the sample temperature (T.sub.2). The
2nd order polynomial curve was used to fit the plot and the a1 and
a2 constants for that level of glucose were obtained as provided in
Table 4. For example, a computer program such as Slidewrite by
Advanced Graphics Software Inc., or any other equivalent curve
fitting program can be used.
TABLE 4 Lot C - 2nd Order Polynomial Coefficients 50 100 200 400
600 Coefficient mg/dL mg/dL mg/dL mg/dL mg/dL a0 29.689 68.654
146.318 301.709 457.305 a1 1.08071 1.06138 1.04494 1.00696 0.95187
a2 -0.00881 0.01035 0.04829 0.12417 0.20045 Corr.Coef.R 0.9990
1.000 0.9998 0.9996 0.9995
The a1 values obtained for the different levels of glucose mere
plotted against the glucose concentration. The data was plotted
using a linear fit, and the coefficients S1 (slope of the linear
fit) and I1 (intercept of the linear fit) were generated. The
Slidewrite program on a PC by Advanced Graphics Software Inc., or
any other equivalent curve fitting program can be used.
The a2 values obtained for the different levels of glucose were
also plotted against the glucose concentration. The data was
plotted using a linear fit, and the coefficients S2 (slope of the
linear fit) and I2 (intercept of the linear fit) were
generated.
To derive the algorithm: at each level of glucose, the observed
glucose concentration (G.sub.1) is related to the sample
temperature (T.sub.2) in a 2nd order polynomial relationship.
And at a sample temperature of 24.degree. C., G.sub.2
(Corrected)=G.sub.1 (Observed)
G.sub.2 =(24.sup.2)*a2+24*a1+a0 Or Equation 4
Subtracting equation (3) from equation (2) gives:
From the linear plots generated at steps 4 and 5:
and
Combining equation (5), (6), and (7) gives equation (2).
While the present invention has been described with reference to
the details of the embodiments of the invention shown in the
drawing, these details are not intended to limit the scope of the
invention as claimed in the appended claims.
* * * * *